In a food processor, combine flours and salt. Add butter; pulse until the butter is in pea-sized pieces. Drizzle in water until the dough starts to come together. Rest in the fridge until ready to assemble.

In a Dutch oven, caramelize mushrooms and onions in olive oil until deeply browned. Deglaze with vinegar and then add the garlic; cook until fragrant. Stir in the miso paste; season to taste. Remove from heat and cool completely.

Preheat oven to 400oF. Roll dough into a 14-inch circle and transfer to a baking stone. Spread filling on dough, leaving a 2-inch perimeter uncovered. Fold edges over. Bake for 45 minutes. Remove tart from the oven and turn the broiler on high. Grate cheese directly over the tart then broil until the cheese begins to brown. Once the cheese is browned, remove the galette from the oven and garnish with thyme.